A  METHOD  WHICH  xMAY  HAVE  BEEN 
USED  BY  THE  MAYAS  IN 
CALCULATING  TIME 


BY 


CHARLES  P.  BOWDITCH 


CAMBRIDGE 

THE  UNIVERSITY  PRESS 

■'.*  ■  „  /> Vfl/  ’*  •  .  Vvv  - 

1901 


Sti-HI 


EX-LIBRIS 

RICARDO  DE  ROBINA 


A  METHOD  WHICH  MAY  HAVE  BEEN 
USED  BY  THE  MAYAS  IN 
CALCULATING  TIME 


BY 

CHARLES  P.  BOWDITCH 


CAMBRIDGE 

THE  UNIVERSITY  PRESS 
1901 


Digitized  by  the  Internet  Archive 
in  2018  with  funding  from 
Getty  Research  Institute 


https://archive.org/details/methodwhichmayhaOObowd 


A  METHOD  WHICH  MAY  HAVE  BEEN  USED  BY  THE 
MAYAS  IN  CALCULATING  TIME 


By  CHARLES  P.  BOWDITCH 

It  is  now  considered  settled  that  the  Initial  Series,  so-called, 
of  the  Maya  inscriptions  had  for  the  basis  of  the  long  count  a 
day  4  Ahau  8  Curnhu,  and  that  in  almost  all  cases  by  counting 
forward  from  that  day  the  number  of  days  as  given  in  the  long 
count,  the  day  and  month  is  reached  which  is  given  in  the  glyphs 
following  the  period  glyphs. 

One  method  which  has  been  used  by  Americanists  for  calcu¬ 
lating  the  position  of  this  day  and  month  in  the  Maya  calendar 
is  as  follows : 

Calculate  the  number  of  days  as  given  in  the  long  count. 

Divide  the  result  by  13,  and  add  the  remainder  to  4,  in  order 
to  find  the  number  attached  to  the  day  sign. 

Divide  the  result  by  20  and  count  forward  the  remainder 
from  Ahau,  in  order  to  find  the  day.  This  can  of  course  be 
simplified  by  merely  counting  forward  from  Ahau  the  number 
attached  to  the  kin  sign,  since  all  the  other  periods  are  exactly 
divisible  by  20. 

Divide  the  result  by  365  and  count  forward  the  remainder  from 
8  Curnhu,  in  order  to  find  the  day  of  the  month,  and  month. 

An  example  is  given  from  Stela  25  of  Piedras  Negras.  Here 
the  Initial  Series  is  (using  Goodman’s  notation)  54.9.8.10.6. 
16.,  10  Cib  9  Mac.  This  period  will  then  be 

9  X  144,000  =  1,296,000 

8  x  7,200  —  57,600 

10  X  360  =  3,600 

6  X  20  =  120 

16  X  1  —  16 

6357*336 

L357>336  =  13X104,410+  6  6  +  4=  10 

=  20  X  67,866  +  16  Ahau  +  16  =  Cib. 

=  365  X  3,718  +  266  8  Curnhu  +  266  =  9  Mac. 


2 


METHODS  WHICH  MAY  HAVE  BEEN  USED  BY 


This  is  a  very  tedious  method  and  has  been  simplified  by 
the  Tables  of  Goodman  and  Gates;  but  there  is  nothing  to  show 
that  the  Mayas  had  such  Tables  or  that  they  adopted  such  a 
long  process  in  order  to  reach  their  results.  We  do  not  know 
that  they  were  able  to  perform  long  division,  and  even  if  they 
knew  how  to  accomplish  this  feat,  it  could  not  have  failed  to 
have  been  a  very  complicated  matter  with  their  vigesimal  sys¬ 
tem,  especially  as  this  system  was  not  regular  in  the  uinal  term. 
Such  divisions  as  i, 357*336  by  365,  for  instance,  would  have  been 
avoided  in  all  probability  if  a  simpler  way  could  have  been  found. 
The  following  method  is  suggested  as  one  which  may  have  been 
in  use  among  them.  I  use  the  plus  sign  to  mean  “  count  for¬ 
ward  ”  and  the  minus  sign  to  mean  “count  backward.” 

i°.  To  find  the  number  attached  to  tJie  day  sign. 

To  the  number  attached  to  the  cycle  sign,  add  twice  the 
number  attached  to  the  katun  sign  and  four  times  the  number 
attached  to  the  tun  sign,  giving  the  sum  the  minus  sign. 

To  seven  times  the  number  attached  to  the  uinal  sign,  add  the 
number  attached  to  the  kin  sign,  giving  this  sum  the  plus  sign. 

Add  these  sums  together,  adding  the  result  to  4  (of  4  Ahau) 
if  the  sign  is  plus,  or  subtracting  it  from  4  if  the  sign  is  minus. 
In  the  first  case  subtract  as  many  times  13  as  possible  from  the 
last  sum.  I11  the  second  case,  add  to  4  as  many  times  13  as 
are  needed  to  make  it  larger  than  the  number  to  be  subtracted 
from  it.  The  result  is  the  number  attached  to  the  day  sign 
sought  for. 

2°.  To  find  the  day  sign. 

Count  forward  from  Ahau  the  number  of  days  equal  to  the 
number  attached  to  the  kin  sign.  The  result  is  the  day  sign 
sought  for. 

30.  To  find  the  month  and  month  number. 

Give  the  plus  sign  to  the  long  count. 

Give  the  minus  sign  to  the  cycle,  katun,  and  tun  count 
of  9.9.16. 


THE  MAYAS  IN  CALCULATING  TIME 


3 


Add  the  two  together  and  give  the  sum  the  proper  sign. 

Multiply  the  resulting  katun  number  by  ioo  and  the  tun 
number  by  5,  add  the  two  products,  and  give  the  sum  the  sign 
opposite  to  the  resulting  sign. 

Multiply  the  resulting  uinal  number  by  20  and  to  the  product 
add  the  kin  sign,  giving  the  sum  the  sign  as  given. 

Add  the  last  two  sums  together,  giving  the  final  sum  the 
proper  sign,  and  after  deducting  as  many  times  365  as  possible 
count  backwards  or  forwards  the  remainder  from  8  Cumhu 
according  to  the  sign. 

It  is  not  impossible  that  this  is  the  method  suggested  on 
page  24  of  the  Dresden  Codex. 

For  30,  the  following  method  can  be  substituted,  which  is 
somewhat  simpler  in  the  use  of  the  plus  and  minus  signs. 

3°  {bis). 

Give  the  minus  sign  to  the  long  count,  omitting  the  uinals 
and  kins. 

Give  the  plus  sign  to  the  cycle,  katun,  and  tun  count  of 
9.9.  16. 

Add  the  two  together  and  give  the  sum  the  proper  sign. 

Multiply  the  katun  number  of  this  sum  by  100  and  the  tun 
number  by  5  and  add  them  together  with  the  proper  sign. 

Multiply  the  uinal  of  the  long  count  by  20  and  add  the 
product  to  the  kin  number  with  the  plus  sign. 

Add  these  last  two  sums  together  with  the  proper  sign. 

After  deducting  as  many  times  365  as  possible  from  this 
last  sum,  count  backwards  or  forwards  the  remainder  from  8 
Cumhu  according  to  the  sign. 

I  give  here  three  examples  from  the  Inscriptions  of  these 
methods  of  calculation. 

Stela  25  of  Piedras  Negras  has  the  Initial  Series  54.9.8. 
10.6. 16.,  10  Cib  9  Mac. 


4 


METHODS  WHICH  MAY  HAVE  BEEN  USED  BY 


i°.  9X1=9  6  X  7  =  42 

8  X  2  =  16  16 

10  X  4  =  40 

-  65  +58 

+  58 


—  7 

4+I3  =  I7“~7  — 
20.  Ahau  +  16  = 

30.  —  9.9.16.0.  o. 

+  9.8.10.6.16. 

—  1.  5. 11. 4. 


I 

X 

100 

=  + 

0 

0 

5 

X 

s 

=  + 

25 

+  125 

1 1 

X 

20 

=  — 

220 

— 

4 

—  224 

“  99 

8  Cuinhu  —  99  = 

3°  (bis). 

+  9.9.16. 

—  9.8.10. 

+  i-6 

1  x  100  =  +  100  6  X  20  =  4-  120 

6  X  s  =  +  3°  +16 

+  130  +  136 

+  136 

+  266 

8  Cumhu  +  266  = 

Stela  36  of  Piedras  Negras  has 
6.5.9.,  8  Muluc  2  Zip. 

i°,  9X1=9 

10  x  2  =  20 
6  x  4  =  24 

—  S3  +44 

+  44 


the  Initial  Series 

5  X  7  =  35 
9 


—  9 

4  +  13  =  *7  “  9  = 


10 

Cib. 


9  Mac. 


9  Mac. 
54-9-IO. 


8 


THE  MAYAS  IN  CALCULATING  TIME 


5 


20.  Ahau  +  9  = 

3°.  —  9.  9.16.0.0. 

+  9.10.  6.5.9. 

+  10.5.9. 

10  X  5  =  —  5° 

5  X  20=  |  100 

+9  +  109 

+  59 

8  Cumhu  +  59  — 

3°  (to). 

+  9.  9.16. 

—  9.10.  6. 


—  0.10. 

10  X  5  =  —  5° 

5  X  20  =  +  100 

+  9  +109 

+  59  as  before. 

On  Stela  B.  of  Copan  is  the  Initial  Series  54.9.  15 .0.0.0., 
Ahau  13  Yax. 

i°.  9X1=9 

15X2  =  3° 

-  39  4  +  (3  X  13)  =  43  -  39  =  4 

20.  Ahau  +  o  =  Ahau. 

30.  +  9.15.  0.0.0. 

—  9.  9.16.0.0. 

+  5.  4.0.0. 

5  X  100  =  —  500 

4  X  5  =  —  20 

—  520 

365 


-  r55 

8  Cumhu  —  155  =  13  Yax. 

3°  (to). 

—  9.15.  o. 

+  9.  9.16. 


-  5-  4- 

5  X  100  =  —  500  etc.  as  before. 

All  these  calculations  can  easily  be  done  in  the  head,  and  the 


Muluc. 


2  Zip. 


6 


METHODS  WHICH  MAY  HAVE  BEEN  USED  BY 


principle  can  be  applied  to  smaller  numbers  than  those  found  in 
the  Initial  Series. 

It  may  be  well  to  add  here  a  fourth  example  from  page 
24  of  the  Dresden  Codex.  Here  we  have  the  two  long  counts 

9.9.16.  0.0. 

9.9.  9.16.0. 

Taking  the  latter  count,  we  find 

i°.  9X1=  9  16  X  7  =  +  112 

9X2=  18 

9X4=  36 


-  63 
+  112 


+  49 

(3  x  13)  39 

0 

+ 

4  +  10  =  14  —  13  =  1 

20.  Ahau  +  0  = 

Ahau. 

30.  — 9.9.16.  0.0. 

(found  on  page  24) 

+  9.9.  9.16.0. 

((  U 

—  6.  2.0. 

(c  a 

6  X  s  =  +  30 
2  X  20  =  —  40 


—  10  8  Cumhu  — 

■  10  —  18  Kayab. 

and  1  Ahau  18  Kayab  is  found  under 

the  second  column 

from  the  left. 

3°  (£")• 

+  9.9.16.  0.0. 

-  9-9-  9- 

+  7- 

7  X  5  =  +  35 

16  X  20  =  +  320 

+  355 

8  Cumhu  +  355  =  18  Kayab  as  before. 


The  reason  why  these  results  are  produced  is  not  far  to  seek. 


THE  MAYAS  IN  CALCULATING  TIME 


7 


i°.  Each  cycle  consists  of  144,000  days.  This  number  is 
divisible  by  13  with  a  remainder  of  12.  If  then  we  count 

forward  one  cycle  from  a  given  day,  say  4  Ahau,  the  number 

of  the  day  will  be  4  +  12  =  16  —  13  =  3.  The  same  result  will 
be  produced,  if,  instead  of  adding  12,  we  should  subtract  1. 
To  find  the  day-number  after  adding  two  cycles,  we  must  add 
twice  12  or  subtract  twice  1,  and  so  on  for  any  number  of 
cycles. 

In  the  same  way  a  katun  is  7200  days.  This  number  is 
divisible  by  13  with  a  remainder  of  11.  If  then  we  wish  to 
count  forward  a  katun  from  a  given  day,  we  can  find  the 

number  of  that  day  by  adding  11  or  by  subtracting  2.  And 

so  on  for  any  number  of  katuns. 

The  360  days  of  a  tun  is  divisible  by  13  with  a  remainder 
of  9.  If  we  wish  to  count  forward  from  a  given  day  one  tun, 

we  shall  find  the  number  of  the  day  sought  by  adding  9  to  the 

number  of  the  given  day  or  by  subtracting  4.  And  so  for  any 
number  of  tuns.  In  each  of  these  three  cases  we  subtract, 
so  that  we  may  use  smaller  numbers  in  our  calculations. 

But  with  the  uinal,  we  find  that  20  is  divisible  by  13  with 
a  remainder  of  7.  So  in  counting  forward  one  uinal,  we  can 
add  7  or  subtract  6.  There  is  not  much  choice  here,  and  so 
we  choose  to  add,  as  we  do  also  with  the  kin  number. 

2°.  As  has  been  already  explained,  the  day  sought  is  found  by 
counting  forward  the  number  of  the  kin  sign  from  the  day  started 
from,  because  all  the  other  period  numbers  are  divisible  by  20, 
the  number  of  days  in  a  day  round. 

3°.  Here  our  object  is  to  count  forward  the  number  of  days 
given  in  the  Initial  Series.  But  in  order  to  bring  this  count  to 

lower  terms  without  altering  its  value,  we  subtract  (or  give  the 

minus  sign  to)  a  period  of  72  calendar  rounds,  or  9 . 9 . 16  .,  where- 
ever,  as  is  almost  always  the  case  in  the  Initial  Series,  the  cycle 
number  is  a  9.  Such  a  subtraction  makes  no  difference  in  the 
final  count. 


8 


METHODS  WHICH  MAY  HAVE  BEEN  USED  BY 


Taking  the  case  of  Stela  36  of  Piedras  Negras,  we  find  the 
first  sum  to  be  -f  10.5.9.  We  have  therefore  to  count  forwards 
10  tuns,  5  uinals,  and  9  kins.  Taking  up  the  tuns  at  first,  we  can 
call  these  ten  years,  which  will  bring  us  to  the  same  month  and 
day  of  the  month  with  which  we  started,  provided  we  count  back 
five  days  for  each  year.  That  is  we  must  count  back  5  X  10  days 
or  —  50.  But  we  must  also  count  forward  the  number  of  days 
represented  by  the  uinals  and  kins,  which  gives 

5  X  20  =  +  100 
+  9 

+  109 

Deducting  the  50  days  obtained  from  the  tuns,  we  have  a  remainder 
of  +  59,  which  being  counted  forward  from  8  Cumhu,  brings  us  to 
2  Zip. 

30  {bis).  Here  we  have  pursued  exactly  the  same  method 
except  that  we  have  changed  the  signs  of  the  long  count  and  of 
the  9.9. 16.,  in  order  not  to  have  so  much  changing  of  signs  in 
the  later  calculations. 

It  should  also  be  said  that  the  count  of  9 . 9 . 16  .  is  not  the  only 
one  which  can  be  used  in  order  to  bring  lower  terms  into  use. 
Any  number  of  complete  calendar  rounds  will  answer  the  pur¬ 
pose,  though,  as  the  count  of  9 . 9 . 16 .  is  given  on  page  24  of  the 
Dresden  Codex,  and  as  most  ‘of  the  cycle  numbers  in  the  Initial 
Series  are  9’s,  it  has  been  taken  as  a  convenient  number.  A  com¬ 
plete  list  of  calendar  rounds  from  one  to  eighty,  in  the  notations 
of  cycles,  katuns,  tuns,  and  uinals,  is  given  in  the  Table. 

But  probably  the  Mayas  would  wish  to  make  this  long  count 
coincide  with  the  count  which  is  given  in  the  Books  of  Chilan 
Balam,  where  the  count  is  by  katuns  which  are  designated  as  Katun 
13  Ahau,  Katun  11  Ahau,  Katun,  9  Ahau,  etc.  This  is  easily  done 
by  adding  the  cycle  number  to  twice  the  katun  number,  deducting 
as  many  thirteens  as  possible  and  subtracting  the  remainder  from 
4  or  from  4  increased  by  13  or  a  multiple  of  13. 


THE  MAYAS  IN  CALCULATING  TIME 


9 


Thus  in  the  first  example,  54.9.8.  10 . 6 .  16 .,  we  take 

9Xi=—  9 
8X2-I - 16 

-  25 

!3 

—  12 

and  4  +  13  —  12  =  5,  and  Katun  5  Ahau  is  the  katun  in  which 
this  date  appears,  or,  in  other  words,  5  Ahau  is  the  beginning 
day  of  54.9.8.0.0.0. 

Again  the  second  example  is  54.9.10.6.5.9.  Here  we 
have 

9  X  1  =  —  9 
10X2  =  —  20 

-  29 

Deduct  twice  thirteen,  or  26 

-  3- 

4  —  3  =  1,  and  Katun  1  Ahau  is  the  katun  in  which  the  date 
appears,  or  1  Ahau  is  the  beginning  day  of  54.9.10.0.0.0. 

In  the  third  example  we  have  54 . 9 .  1 5  .  o .  o  .  o  .,  and  the  same 
calculation  brings 

9  X  1  =  —  9 
15  x  2  =  —  30 

-  39 

Deduct  three  times  thirteen,  or  39 

o 

4  —  0  =  4,  and  Katun  4  Ahau  is  the  katun  in  which  this  date 
appears,  or  4  Ahau  is  the  beginning  day  of  54.9.  15.0. 0.O. 

And  finally  on  page  24  of  the  Dresden  Codex,  where  however 
there  is  no  large  glyph  at  the  beginning  of  the  Series  as  there  is 


IO 


METHODS  WHICH  MAY  HAVE  BEEN  USED  BY 


invariably  in  the  Inscriptions,  we  have  9.9.9.16.0.  Here  the 
calculation  is 


9Xi=- 9 
9  x  2  =  —  18 

_ — 

Z/ 

Deduct  twice  thirteen,  or  26 

—  1 

4  —  1=3,  and  Katun  3  Ahau  is  the  katun  in  which  this  date 
appears. 


I 

2 

3 

4 

5 

6 

7 

8 

9 

io 

ii 

12 

13 

14 

15 

l6 

17 

l8 

19 

20 

21 

22 

23 

24 

25 

26 

27 

28 

29 

3° 

31 

32 

33 

34 

35 

36 

37 

38 

39 

40 


THE  MAYAS  IN  CALCULATING  TIME 


II 


Days. 


18,980 

37,960 

5^,94° 

75,920 

94,900 

113,880 

132,860 

151,840 

170,820 

189,800 

208,780 

227,760 

246,740 

265,720 

284,700 

303,680 

322,660 

341,640 

360,620 

379,600 

398,580 

417,560 

436,54° 

455,520 

474,5°° 

493,48o 

512,460 

53  r> 44° 
550,420 
569,400 
588,380 
607,360 
626,340 
645,320 
664,300 
683,280 
702,260 
721,240 
740,220 
759,200 


TABLE. 


Cycles,  etc. 

Calendar 

Rounds. 

Days. 

Cycles,  etc. 

2.12.13.0. 

4i 

778,180 

5.  8.  1.11.0. 

5.  5.  8.0. 

42 

797,160 

5. 10.14.  6.0. 

7.18.  3.0. 

43 

816,140 

5  !3-  7-  10. 

10.10.16.0. 

44 

835,120 

5.15.19.14.0. 

13.  3.11.0. 

45 

854,100 

5.18.12.  9.0. 

15.16.  6.0. 

46 

873,080 

6.  1.  5.  4.0. 

18.  9.  1.0. 

47 

892,060 

6.  3.17.17.0. 

I.  I.  1. 14.0. 

48 

911,040 

6.  6.10.12.0. 

1.  3.14.  9.0. 

49 

930,020 

6.  9  3.  7.0. 

1.  6.  7.  4.0. 

5° 

949,000 

6.11.16.  2.0. 

1.  8.19.17.0. 

5i 

967,980 

6.14.  8.15.0. 

1.11.12.12.0. 

52 

986,960 

6.17.  1. 100. 

1. 14.  5.  7.0. 

53 

1,005,940 

6.19.14.  5.0. 

1. 16.18.  2.0. 

54 

1,024,920 

7.  2.  7.  0.0. 

1.19.10.15  0. 

55 

1,043,900 

7.  4.19.13.0. 

2.  2.  3.10.0. 

56 

1,062,880 

7.  7.12.  8.0. 

2.  4.16.  50. 

57 

1,081,860 

7.10.  5.  3.0. 

2.  7.  9.  0.0. 

58 

1,100,840 

7.12.17.16.0. 

2.  ia.  1. 13.0. 

59 

1,119,820 

7.15.10.11.0. 

2. 12. 14.  8.0. 

60 

1,138,800 

7.18.  3.  6.0. 

2.15.  7.  3.0. 

61 

1,157,780 

8.  0.16.  1.0. 

2.17.19.16.0. 

62 

1,176,760 

8.  3.  8.14.0. 

3.  0.12.11.0. 

63 

1,195,740 

8.  6.  1.  9.0. 

3.  3.  5.  6.0. 

64 

1,214,720 

8.  8.14.  4.0. 

3.  5.18.  1 .0. 

65 

1,233,700 

8. 11.  6.17.0. 

3.  8.10.140. 

66 

1,252,680 

8.13.19.12.0. 

3.U.  3.  9.0. 

67 

1,271,660 

8.16.12.  7.0. 

3.13.16.  4.0. 

68 

1,290,640 

8.19.  5.  2.0. 

3.16.  8.17.0. 

69 

1,309,620 

9.  1.17.15.0. 

3.19.  1. 12.0. 

70 

1,328,600 

9.  4.10.10.0. 

4.  1. 14.  7.0. 

7i 

i,347,58o 

9.  7.  3.  5.0. 

4.  4.  7.  2.0. 

72 

1,366,560 

9.  9.16.  0.0. 

4.  6.19.15.0. 

73 

1,385,540 

9.12.  8.13.0. 

4.  9.12. IO.O. 

74 

1,404,520 

9.15.  1.  8.0. 

4.12.  5.  5.0. 

75 

1,423,500 

9-I7-I4-  3°- 

4.14.18.  0.0. 

76 

1,442,480 

10.  0.  6.16.0. 

4.17.10.13.0. 

77 

1,461,460 

10.  2. 1 9. 1 1. 0. 

5.  0.  3.  8.0. 

78 

1,480,440 

10.  5.12.  6.0. 

5.  2.16.  3.0. 

79 

1,499,420 

10.  8.  5.  1.0. 

5.  5.  8.16.0. 

80 

1,518,400 

10.10.17.14.0. 

1  o  i 


